Real-rooted atlas

A062145 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 + 18\,x - 8\,x^{2} - 7\,n + 15\,n x + 12\,n x^{2} - 3\,n^{2} - 33\,n^{2} x + 6\,n^{2} x^{2} + 8\,n^{3} - 18\,n^{3} x - 11\,n^{3} x^{2} + 4\,n^{4} + 12\,n^{4} x - 3\,n^{4} x^{2} + 6\,n^{5} x + 3\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(8\,x - 22\,x^{2} + 6\,x^{3} + 2\,n + 18\,n x + 23\,n x^{2} - 23\,n x^{3} + 12\,n^{2} + 58\,n^{2} x^{2} + 36\,n^{2} x^{3} + 24\,n^{3} + 28\,n^{3} x - 10\,n^{3} x^{2} + 3\,n^{3} x^{3} + 16\,n^{4} + 84\,n^{4} x + 69\,n^{4} x^{2} + 14\,n^{4} x^{3} + 24\,n^{5} x + 40\,n^{5} x^{2} + 15\,n^{5} x^{3} + 4\,n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-x - 3\,x^{2} + x^{3} + 3\,x^{4} - 6\,n x - 2\,n x^{2} - 6\,n x^{3} - 10\,n x^{4} - 12\,n^{2} x - 20\,n^{2} x^{2} - 3\,n^{2} x^{3} + 5\,n^{2} x^{4} - 8\,n^{3} x - 44\,n^{3} x^{2} - 48\,n^{3} x^{3} - 12\,n^{3} x^{4} - 12\,n^{4} x^{2} - 23\,n^{4} x^{3} - 11\,n^{4} x^{4} - 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
251621
32112090231
4846729453202021
533033606804448087515849
61287158404158040320157502016116793
7500572072231660295680173250453604116827143
81944832032012162151921920150150059875211319076805699025

All 15 rows

Technical details
Node id
n-fecfb9956dc433a3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b6f4e21d980b6453

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      5,      16;
   3:     21,     120,       90;
   4:     84,     672,      945,      320;
   5:    330,    3360,     6804,     4480,      875;
   6:   1287,   15840,    41580,    40320,    15750,     2016;
   7:   5005,   72072,   231660,   295680,   173250,    45360,     4116;
   8:  19448,  320320,  1216215,  1921920,  1501500,   598752,   113190,    7680;
   9:  75582, 1400256,  6126120, 11531520, 11261250,  6054048,  1765764,  253440,  13365;
  10: 293930, 6046560, 29930472, 65345280, 76576500, 51891840, 20600580, 4612608, 521235, 22000;
%F Formula
The row polynomials satisfy (4*n^4+8*n^3-3*n^2-7*n-2+(6*n^5+12*n^4-18*n^3-33*n^2+15*n+18)*x+(n^6+3*n^5-3*n^4-11*n^3+6*n^2+12*n-8)*x^2)*P(n,x) = (16*n^4+24*n^3+12*n^2+2*n+(24*n^5+84*n^4+28*n^3+18*n+8)*x+(4*n^6+40*n^5+69*n^4-10*n^3+58*n^2+23*n-22)*x^2+(3*n^6+15*n^5+14*n^4+3*n^3+36*n^2-23*n+6)*x^3)*P(n-1,x) - ((8*n^3+12*n^2+6*n+1)*x+(12*n^4+44*n^3+20*n^2+2*n+3)*x^2+(2*n^5+23*n^4+48*n^3+3*n^2+6*n-1)*x^3+(2*n^5+11*n^4+12*n^3-5*n^2+10*n-3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^4 + 24*n^3 + 12*n^2 + 2*n + (24*n^5 + 84*n^4 + 28*n^3 + 18*n + 8)*x + (4*n^6 + 40*n^5 + 69*n^4 - 10*n^3 + 58*n^2 + 23*n - 22)*x^2 + (3*n^6 + 15*n^5 + 14*n^4 + 3*n^3 + 36*n^2 - 23*n + 6)*x^3)*P[n - 1, x] - ((8*n^3 + 12*n^2 + 6*n + 1)*x + (12*n^4 + 44*n^3 + 20*n^2 + 2*n + 3)*x^2 + (2*n^5 + 23*n^4 + 48*n^3 + 3*n^2 + 6*n - 1)*x^3 + (2*n^5 + 11*n^4 + 12*n^3 - 5*n^2 + 10*n - 3)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 8*n^3 - 3*n^2 - 7*n - 2 + (6*n^5 + 12*n^4 - 18*n^3 - 33*n^2 + 15*n + 18)*x + (n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 6*n^2 + 12*n - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n^4 + 24*n^3 + 12*n^2 + 2*n + (24*n^5 + 84*n^4 + 28*n^3 + 18*n + 8)*x + (4*n^6 + 40*n^5 + 69*n^4 - 10*n^3 + 58*n^2 + 23*n - 22)*x^2 + (3*n^6 + 15*n^5 + 14*n^4 + 3*n^3 + 36*n^2 - 23*n + 6)*x^3)*v[i - 1] - ((8*n^3 + 12*n^2 + 6*n + 1)*x + (12*n^4 + 44*n^3 + 20*n^2 + 2*n + 3)*x^2 + (2*n^5 + 23*n^4 + 48*n^3 + 3*n^2 + 6*n - 1)*x^3 + (2*n^5 + 11*n^4 + 12*n^3 - 5*n^2 + 10*n - 3)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 8*n^3 - 3*n^2 - 7*n - 2 + (6*n^5 + 12*n^4 - 18*n^3 - 33*n^2 + 15*n + 18)*x + (n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 6*n^2 + 12*n - 8)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.